Properties

Label 2151.2.a.k.1.17
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Root \(2.39589\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.39589 q^{2} +3.74030 q^{4} +4.10699 q^{5} -3.62464 q^{7} +4.16958 q^{8} +O(q^{10})\) \(q+2.39589 q^{2} +3.74030 q^{4} +4.10699 q^{5} -3.62464 q^{7} +4.16958 q^{8} +9.83991 q^{10} +1.86664 q^{11} +3.35969 q^{13} -8.68425 q^{14} +2.50926 q^{16} +0.173143 q^{17} -1.09786 q^{19} +15.3614 q^{20} +4.47226 q^{22} -5.06374 q^{23} +11.8674 q^{25} +8.04946 q^{26} -13.5573 q^{28} +6.94337 q^{29} +6.11145 q^{31} -2.32723 q^{32} +0.414831 q^{34} -14.8864 q^{35} +3.44771 q^{37} -2.63035 q^{38} +17.1244 q^{40} -5.66600 q^{41} -5.01704 q^{43} +6.98178 q^{44} -12.1322 q^{46} -5.07918 q^{47} +6.13803 q^{49} +28.4330 q^{50} +12.5663 q^{52} -4.55248 q^{53} +7.66626 q^{55} -15.1132 q^{56} +16.6356 q^{58} -4.23002 q^{59} +8.14825 q^{61} +14.6424 q^{62} -10.5943 q^{64} +13.7982 q^{65} -8.96008 q^{67} +0.647606 q^{68} -35.6661 q^{70} +13.9037 q^{71} -6.34022 q^{73} +8.26034 q^{74} -4.10632 q^{76} -6.76589 q^{77} +1.38423 q^{79} +10.3055 q^{80} -13.5751 q^{82} -14.9390 q^{83} +0.711095 q^{85} -12.0203 q^{86} +7.78309 q^{88} +16.3643 q^{89} -12.1777 q^{91} -18.9399 q^{92} -12.1692 q^{94} -4.50888 q^{95} -17.5839 q^{97} +14.7061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.39589 1.69415 0.847076 0.531472i \(-0.178361\pi\)
0.847076 + 0.531472i \(0.178361\pi\)
\(3\) 0 0
\(4\) 3.74030 1.87015
\(5\) 4.10699 1.83670 0.918351 0.395767i \(-0.129521\pi\)
0.918351 + 0.395767i \(0.129521\pi\)
\(6\) 0 0
\(7\) −3.62464 −1.36999 −0.684993 0.728550i \(-0.740195\pi\)
−0.684993 + 0.728550i \(0.740195\pi\)
\(8\) 4.16958 1.47417
\(9\) 0 0
\(10\) 9.83991 3.11165
\(11\) 1.86664 0.562812 0.281406 0.959589i \(-0.409199\pi\)
0.281406 + 0.959589i \(0.409199\pi\)
\(12\) 0 0
\(13\) 3.35969 0.931811 0.465905 0.884835i \(-0.345729\pi\)
0.465905 + 0.884835i \(0.345729\pi\)
\(14\) −8.68425 −2.32096
\(15\) 0 0
\(16\) 2.50926 0.627316
\(17\) 0.173143 0.0419933 0.0209966 0.999780i \(-0.493316\pi\)
0.0209966 + 0.999780i \(0.493316\pi\)
\(18\) 0 0
\(19\) −1.09786 −0.251865 −0.125933 0.992039i \(-0.540192\pi\)
−0.125933 + 0.992039i \(0.540192\pi\)
\(20\) 15.3614 3.43491
\(21\) 0 0
\(22\) 4.47226 0.953489
\(23\) −5.06374 −1.05586 −0.527932 0.849287i \(-0.677032\pi\)
−0.527932 + 0.849287i \(0.677032\pi\)
\(24\) 0 0
\(25\) 11.8674 2.37347
\(26\) 8.04946 1.57863
\(27\) 0 0
\(28\) −13.5573 −2.56208
\(29\) 6.94337 1.28935 0.644676 0.764456i \(-0.276992\pi\)
0.644676 + 0.764456i \(0.276992\pi\)
\(30\) 0 0
\(31\) 6.11145 1.09765 0.548824 0.835938i \(-0.315076\pi\)
0.548824 + 0.835938i \(0.315076\pi\)
\(32\) −2.32723 −0.411401
\(33\) 0 0
\(34\) 0.414831 0.0711430
\(35\) −14.8864 −2.51626
\(36\) 0 0
\(37\) 3.44771 0.566800 0.283400 0.959002i \(-0.408538\pi\)
0.283400 + 0.959002i \(0.408538\pi\)
\(38\) −2.63035 −0.426698
\(39\) 0 0
\(40\) 17.1244 2.70761
\(41\) −5.66600 −0.884880 −0.442440 0.896798i \(-0.645887\pi\)
−0.442440 + 0.896798i \(0.645887\pi\)
\(42\) 0 0
\(43\) −5.01704 −0.765091 −0.382545 0.923937i \(-0.624952\pi\)
−0.382545 + 0.923937i \(0.624952\pi\)
\(44\) 6.98178 1.05254
\(45\) 0 0
\(46\) −12.1322 −1.78879
\(47\) −5.07918 −0.740875 −0.370437 0.928857i \(-0.620792\pi\)
−0.370437 + 0.928857i \(0.620792\pi\)
\(48\) 0 0
\(49\) 6.13803 0.876861
\(50\) 28.4330 4.02103
\(51\) 0 0
\(52\) 12.5663 1.74263
\(53\) −4.55248 −0.625331 −0.312666 0.949863i \(-0.601222\pi\)
−0.312666 + 0.949863i \(0.601222\pi\)
\(54\) 0 0
\(55\) 7.66626 1.03372
\(56\) −15.1132 −2.01959
\(57\) 0 0
\(58\) 16.6356 2.18436
\(59\) −4.23002 −0.550701 −0.275351 0.961344i \(-0.588794\pi\)
−0.275351 + 0.961344i \(0.588794\pi\)
\(60\) 0 0
\(61\) 8.14825 1.04328 0.521639 0.853167i \(-0.325321\pi\)
0.521639 + 0.853167i \(0.325321\pi\)
\(62\) 14.6424 1.85958
\(63\) 0 0
\(64\) −10.5943 −1.32429
\(65\) 13.7982 1.71146
\(66\) 0 0
\(67\) −8.96008 −1.09465 −0.547324 0.836921i \(-0.684353\pi\)
−0.547324 + 0.836921i \(0.684353\pi\)
\(68\) 0.647606 0.0785338
\(69\) 0 0
\(70\) −35.6661 −4.26292
\(71\) 13.9037 1.65006 0.825032 0.565086i \(-0.191157\pi\)
0.825032 + 0.565086i \(0.191157\pi\)
\(72\) 0 0
\(73\) −6.34022 −0.742066 −0.371033 0.928620i \(-0.620996\pi\)
−0.371033 + 0.928620i \(0.620996\pi\)
\(74\) 8.26034 0.960245
\(75\) 0 0
\(76\) −4.10632 −0.471027
\(77\) −6.76589 −0.771044
\(78\) 0 0
\(79\) 1.38423 0.155738 0.0778688 0.996964i \(-0.475188\pi\)
0.0778688 + 0.996964i \(0.475188\pi\)
\(80\) 10.3055 1.15219
\(81\) 0 0
\(82\) −13.5751 −1.49912
\(83\) −14.9390 −1.63977 −0.819886 0.572527i \(-0.805963\pi\)
−0.819886 + 0.572527i \(0.805963\pi\)
\(84\) 0 0
\(85\) 0.711095 0.0771291
\(86\) −12.0203 −1.29618
\(87\) 0 0
\(88\) 7.78309 0.829680
\(89\) 16.3643 1.73461 0.867307 0.497774i \(-0.165849\pi\)
0.867307 + 0.497774i \(0.165849\pi\)
\(90\) 0 0
\(91\) −12.1777 −1.27657
\(92\) −18.9399 −1.97463
\(93\) 0 0
\(94\) −12.1692 −1.25515
\(95\) −4.50888 −0.462602
\(96\) 0 0
\(97\) −17.5839 −1.78537 −0.892687 0.450677i \(-0.851183\pi\)
−0.892687 + 0.450677i \(0.851183\pi\)
\(98\) 14.7061 1.48554
\(99\) 0 0
\(100\) 44.3876 4.43876
\(101\) −9.02895 −0.898414 −0.449207 0.893428i \(-0.648293\pi\)
−0.449207 + 0.893428i \(0.648293\pi\)
\(102\) 0 0
\(103\) 2.28713 0.225358 0.112679 0.993631i \(-0.464057\pi\)
0.112679 + 0.993631i \(0.464057\pi\)
\(104\) 14.0085 1.37365
\(105\) 0 0
\(106\) −10.9073 −1.05941
\(107\) −9.34465 −0.903381 −0.451691 0.892175i \(-0.649179\pi\)
−0.451691 + 0.892175i \(0.649179\pi\)
\(108\) 0 0
\(109\) −0.314568 −0.0301302 −0.0150651 0.999887i \(-0.504796\pi\)
−0.0150651 + 0.999887i \(0.504796\pi\)
\(110\) 18.3675 1.75128
\(111\) 0 0
\(112\) −9.09518 −0.859414
\(113\) 2.33072 0.219255 0.109628 0.993973i \(-0.465034\pi\)
0.109628 + 0.993973i \(0.465034\pi\)
\(114\) 0 0
\(115\) −20.7967 −1.93931
\(116\) 25.9703 2.41128
\(117\) 0 0
\(118\) −10.1347 −0.932972
\(119\) −0.627580 −0.0575302
\(120\) 0 0
\(121\) −7.51567 −0.683243
\(122\) 19.5223 1.76747
\(123\) 0 0
\(124\) 22.8587 2.05277
\(125\) 28.2042 2.52266
\(126\) 0 0
\(127\) −0.790729 −0.0701658 −0.0350829 0.999384i \(-0.511170\pi\)
−0.0350829 + 0.999384i \(0.511170\pi\)
\(128\) −20.7284 −1.83215
\(129\) 0 0
\(130\) 33.0591 2.89947
\(131\) −16.5237 −1.44368 −0.721842 0.692058i \(-0.756704\pi\)
−0.721842 + 0.692058i \(0.756704\pi\)
\(132\) 0 0
\(133\) 3.97933 0.345052
\(134\) −21.4674 −1.85450
\(135\) 0 0
\(136\) 0.721932 0.0619052
\(137\) −15.7754 −1.34778 −0.673891 0.738831i \(-0.735378\pi\)
−0.673891 + 0.738831i \(0.735378\pi\)
\(138\) 0 0
\(139\) 14.9098 1.26463 0.632317 0.774710i \(-0.282104\pi\)
0.632317 + 0.774710i \(0.282104\pi\)
\(140\) −55.6795 −4.70578
\(141\) 0 0
\(142\) 33.3118 2.79546
\(143\) 6.27132 0.524434
\(144\) 0 0
\(145\) 28.5164 2.36815
\(146\) −15.1905 −1.25717
\(147\) 0 0
\(148\) 12.8955 1.06000
\(149\) 16.5686 1.35735 0.678677 0.734437i \(-0.262554\pi\)
0.678677 + 0.734437i \(0.262554\pi\)
\(150\) 0 0
\(151\) −11.4842 −0.934575 −0.467288 0.884105i \(-0.654769\pi\)
−0.467288 + 0.884105i \(0.654769\pi\)
\(152\) −4.57760 −0.371292
\(153\) 0 0
\(154\) −16.2103 −1.30627
\(155\) 25.0997 2.01605
\(156\) 0 0
\(157\) −10.5567 −0.842520 −0.421260 0.906940i \(-0.638412\pi\)
−0.421260 + 0.906940i \(0.638412\pi\)
\(158\) 3.31646 0.263843
\(159\) 0 0
\(160\) −9.55793 −0.755621
\(161\) 18.3543 1.44652
\(162\) 0 0
\(163\) 19.0184 1.48963 0.744816 0.667270i \(-0.232537\pi\)
0.744816 + 0.667270i \(0.232537\pi\)
\(164\) −21.1925 −1.65486
\(165\) 0 0
\(166\) −35.7923 −2.77802
\(167\) 1.53829 0.119036 0.0595181 0.998227i \(-0.481044\pi\)
0.0595181 + 0.998227i \(0.481044\pi\)
\(168\) 0 0
\(169\) −1.71247 −0.131729
\(170\) 1.70371 0.130668
\(171\) 0 0
\(172\) −18.7652 −1.43084
\(173\) −16.0374 −1.21930 −0.609650 0.792671i \(-0.708690\pi\)
−0.609650 + 0.792671i \(0.708690\pi\)
\(174\) 0 0
\(175\) −43.0150 −3.25163
\(176\) 4.68388 0.353061
\(177\) 0 0
\(178\) 39.2071 2.93870
\(179\) 8.92422 0.667027 0.333514 0.942745i \(-0.391766\pi\)
0.333514 + 0.942745i \(0.391766\pi\)
\(180\) 0 0
\(181\) 10.5224 0.782125 0.391062 0.920364i \(-0.372108\pi\)
0.391062 + 0.920364i \(0.372108\pi\)
\(182\) −29.1764 −2.16270
\(183\) 0 0
\(184\) −21.1137 −1.55652
\(185\) 14.1597 1.04104
\(186\) 0 0
\(187\) 0.323194 0.0236343
\(188\) −18.9977 −1.38555
\(189\) 0 0
\(190\) −10.8028 −0.783718
\(191\) 21.1875 1.53307 0.766537 0.642200i \(-0.221978\pi\)
0.766537 + 0.642200i \(0.221978\pi\)
\(192\) 0 0
\(193\) −5.55065 −0.399545 −0.199772 0.979842i \(-0.564020\pi\)
−0.199772 + 0.979842i \(0.564020\pi\)
\(194\) −42.1291 −3.02470
\(195\) 0 0
\(196\) 22.9581 1.63986
\(197\) 16.4993 1.17552 0.587761 0.809034i \(-0.300009\pi\)
0.587761 + 0.809034i \(0.300009\pi\)
\(198\) 0 0
\(199\) 20.6187 1.46162 0.730811 0.682580i \(-0.239142\pi\)
0.730811 + 0.682580i \(0.239142\pi\)
\(200\) 49.4820 3.49890
\(201\) 0 0
\(202\) −21.6324 −1.52205
\(203\) −25.1672 −1.76639
\(204\) 0 0
\(205\) −23.2702 −1.62526
\(206\) 5.47973 0.381791
\(207\) 0 0
\(208\) 8.43035 0.584540
\(209\) −2.04930 −0.141753
\(210\) 0 0
\(211\) −28.3629 −1.95258 −0.976290 0.216466i \(-0.930547\pi\)
−0.976290 + 0.216466i \(0.930547\pi\)
\(212\) −17.0277 −1.16946
\(213\) 0 0
\(214\) −22.3888 −1.53047
\(215\) −20.6049 −1.40524
\(216\) 0 0
\(217\) −22.1518 −1.50376
\(218\) −0.753672 −0.0510451
\(219\) 0 0
\(220\) 28.6741 1.93321
\(221\) 0.581706 0.0391298
\(222\) 0 0
\(223\) 2.14038 0.143331 0.0716653 0.997429i \(-0.477169\pi\)
0.0716653 + 0.997429i \(0.477169\pi\)
\(224\) 8.43539 0.563613
\(225\) 0 0
\(226\) 5.58415 0.371452
\(227\) 21.4911 1.42641 0.713207 0.700953i \(-0.247242\pi\)
0.713207 + 0.700953i \(0.247242\pi\)
\(228\) 0 0
\(229\) −17.5694 −1.16102 −0.580511 0.814253i \(-0.697147\pi\)
−0.580511 + 0.814253i \(0.697147\pi\)
\(230\) −49.8268 −3.28548
\(231\) 0 0
\(232\) 28.9510 1.90072
\(233\) −12.7594 −0.835898 −0.417949 0.908470i \(-0.637251\pi\)
−0.417949 + 0.908470i \(0.637251\pi\)
\(234\) 0 0
\(235\) −20.8601 −1.36077
\(236\) −15.8215 −1.02990
\(237\) 0 0
\(238\) −1.50361 −0.0974648
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −12.5216 −0.806589 −0.403294 0.915070i \(-0.632135\pi\)
−0.403294 + 0.915070i \(0.632135\pi\)
\(242\) −18.0067 −1.15752
\(243\) 0 0
\(244\) 30.4769 1.95109
\(245\) 25.2088 1.61053
\(246\) 0 0
\(247\) −3.68846 −0.234691
\(248\) 25.4822 1.61812
\(249\) 0 0
\(250\) 67.5743 4.27378
\(251\) 14.0327 0.885739 0.442869 0.896586i \(-0.353961\pi\)
0.442869 + 0.896586i \(0.353961\pi\)
\(252\) 0 0
\(253\) −9.45216 −0.594252
\(254\) −1.89450 −0.118872
\(255\) 0 0
\(256\) −28.4744 −1.77965
\(257\) 5.67848 0.354214 0.177107 0.984192i \(-0.443326\pi\)
0.177107 + 0.984192i \(0.443326\pi\)
\(258\) 0 0
\(259\) −12.4967 −0.776508
\(260\) 51.6095 3.20069
\(261\) 0 0
\(262\) −39.5890 −2.44582
\(263\) −5.16054 −0.318212 −0.159106 0.987261i \(-0.550861\pi\)
−0.159106 + 0.987261i \(0.550861\pi\)
\(264\) 0 0
\(265\) −18.6970 −1.14855
\(266\) 9.53406 0.584571
\(267\) 0 0
\(268\) −33.5134 −2.04716
\(269\) 1.69229 0.103181 0.0515903 0.998668i \(-0.483571\pi\)
0.0515903 + 0.998668i \(0.483571\pi\)
\(270\) 0 0
\(271\) −11.7243 −0.712203 −0.356101 0.934447i \(-0.615894\pi\)
−0.356101 + 0.934447i \(0.615894\pi\)
\(272\) 0.434461 0.0263430
\(273\) 0 0
\(274\) −37.7961 −2.28335
\(275\) 22.1521 1.33582
\(276\) 0 0
\(277\) 25.2439 1.51676 0.758380 0.651813i \(-0.225991\pi\)
0.758380 + 0.651813i \(0.225991\pi\)
\(278\) 35.7223 2.14248
\(279\) 0 0
\(280\) −62.0699 −3.70939
\(281\) 23.7234 1.41522 0.707609 0.706604i \(-0.249774\pi\)
0.707609 + 0.706604i \(0.249774\pi\)
\(282\) 0 0
\(283\) 28.7226 1.70738 0.853690 0.520781i \(-0.174359\pi\)
0.853690 + 0.520781i \(0.174359\pi\)
\(284\) 52.0040 3.08587
\(285\) 0 0
\(286\) 15.0254 0.888471
\(287\) 20.5372 1.21227
\(288\) 0 0
\(289\) −16.9700 −0.998237
\(290\) 68.3221 4.01201
\(291\) 0 0
\(292\) −23.7143 −1.38778
\(293\) −5.66344 −0.330862 −0.165431 0.986221i \(-0.552901\pi\)
−0.165431 + 0.986221i \(0.552901\pi\)
\(294\) 0 0
\(295\) −17.3726 −1.01147
\(296\) 14.3755 0.835559
\(297\) 0 0
\(298\) 39.6966 2.29956
\(299\) −17.0126 −0.983865
\(300\) 0 0
\(301\) 18.1850 1.04816
\(302\) −27.5150 −1.58331
\(303\) 0 0
\(304\) −2.75481 −0.157999
\(305\) 33.4648 1.91619
\(306\) 0 0
\(307\) −17.7678 −1.01406 −0.507032 0.861927i \(-0.669257\pi\)
−0.507032 + 0.861927i \(0.669257\pi\)
\(308\) −25.3065 −1.44197
\(309\) 0 0
\(310\) 60.1361 3.41550
\(311\) 24.8432 1.40873 0.704365 0.709838i \(-0.251232\pi\)
0.704365 + 0.709838i \(0.251232\pi\)
\(312\) 0 0
\(313\) −29.3512 −1.65903 −0.829514 0.558485i \(-0.811383\pi\)
−0.829514 + 0.558485i \(0.811383\pi\)
\(314\) −25.2928 −1.42736
\(315\) 0 0
\(316\) 5.17742 0.291253
\(317\) −22.7860 −1.27979 −0.639894 0.768463i \(-0.721022\pi\)
−0.639894 + 0.768463i \(0.721022\pi\)
\(318\) 0 0
\(319\) 12.9607 0.725662
\(320\) −43.5108 −2.43233
\(321\) 0 0
\(322\) 43.9748 2.45062
\(323\) −0.190086 −0.0105766
\(324\) 0 0
\(325\) 39.8707 2.21163
\(326\) 45.5659 2.52366
\(327\) 0 0
\(328\) −23.6248 −1.30446
\(329\) 18.4102 1.01499
\(330\) 0 0
\(331\) 23.2727 1.27918 0.639591 0.768715i \(-0.279104\pi\)
0.639591 + 0.768715i \(0.279104\pi\)
\(332\) −55.8765 −3.06662
\(333\) 0 0
\(334\) 3.68557 0.201666
\(335\) −36.7989 −2.01054
\(336\) 0 0
\(337\) 32.4686 1.76868 0.884338 0.466846i \(-0.154610\pi\)
0.884338 + 0.466846i \(0.154610\pi\)
\(338\) −4.10290 −0.223169
\(339\) 0 0
\(340\) 2.65971 0.144243
\(341\) 11.4078 0.617769
\(342\) 0 0
\(343\) 3.12435 0.168699
\(344\) −20.9189 −1.12787
\(345\) 0 0
\(346\) −38.4238 −2.06568
\(347\) −8.61985 −0.462738 −0.231369 0.972866i \(-0.574320\pi\)
−0.231369 + 0.972866i \(0.574320\pi\)
\(348\) 0 0
\(349\) −20.5649 −1.10081 −0.550406 0.834897i \(-0.685527\pi\)
−0.550406 + 0.834897i \(0.685527\pi\)
\(350\) −103.059 −5.50875
\(351\) 0 0
\(352\) −4.34410 −0.231541
\(353\) 28.1526 1.49841 0.749206 0.662337i \(-0.230435\pi\)
0.749206 + 0.662337i \(0.230435\pi\)
\(354\) 0 0
\(355\) 57.1023 3.03068
\(356\) 61.2075 3.24399
\(357\) 0 0
\(358\) 21.3815 1.13005
\(359\) −13.5612 −0.715735 −0.357867 0.933772i \(-0.616496\pi\)
−0.357867 + 0.933772i \(0.616496\pi\)
\(360\) 0 0
\(361\) −17.7947 −0.936564
\(362\) 25.2106 1.32504
\(363\) 0 0
\(364\) −45.5482 −2.38737
\(365\) −26.0392 −1.36295
\(366\) 0 0
\(367\) −25.7987 −1.34668 −0.673340 0.739333i \(-0.735141\pi\)
−0.673340 + 0.739333i \(0.735141\pi\)
\(368\) −12.7063 −0.662360
\(369\) 0 0
\(370\) 33.9252 1.76368
\(371\) 16.5011 0.856695
\(372\) 0 0
\(373\) 35.5762 1.84207 0.921033 0.389484i \(-0.127347\pi\)
0.921033 + 0.389484i \(0.127347\pi\)
\(374\) 0.774339 0.0400401
\(375\) 0 0
\(376\) −21.1781 −1.09217
\(377\) 23.3276 1.20143
\(378\) 0 0
\(379\) −2.52878 −0.129894 −0.0649472 0.997889i \(-0.520688\pi\)
−0.0649472 + 0.997889i \(0.520688\pi\)
\(380\) −16.8646 −0.865136
\(381\) 0 0
\(382\) 50.7630 2.59726
\(383\) −13.7606 −0.703134 −0.351567 0.936163i \(-0.614351\pi\)
−0.351567 + 0.936163i \(0.614351\pi\)
\(384\) 0 0
\(385\) −27.7874 −1.41618
\(386\) −13.2988 −0.676890
\(387\) 0 0
\(388\) −65.7691 −3.33892
\(389\) −23.0695 −1.16967 −0.584836 0.811152i \(-0.698841\pi\)
−0.584836 + 0.811152i \(0.698841\pi\)
\(390\) 0 0
\(391\) −0.876750 −0.0443391
\(392\) 25.5930 1.29264
\(393\) 0 0
\(394\) 39.5304 1.99151
\(395\) 5.68500 0.286043
\(396\) 0 0
\(397\) 16.1634 0.811218 0.405609 0.914047i \(-0.367059\pi\)
0.405609 + 0.914047i \(0.367059\pi\)
\(398\) 49.4003 2.47621
\(399\) 0 0
\(400\) 29.7784 1.48892
\(401\) −13.9016 −0.694215 −0.347107 0.937825i \(-0.612836\pi\)
−0.347107 + 0.937825i \(0.612836\pi\)
\(402\) 0 0
\(403\) 20.5326 1.02280
\(404\) −33.7710 −1.68017
\(405\) 0 0
\(406\) −60.2980 −2.99254
\(407\) 6.43562 0.319002
\(408\) 0 0
\(409\) −7.79717 −0.385545 −0.192773 0.981243i \(-0.561748\pi\)
−0.192773 + 0.981243i \(0.561748\pi\)
\(410\) −55.7529 −2.75344
\(411\) 0 0
\(412\) 8.55457 0.421454
\(413\) 15.3323 0.754453
\(414\) 0 0
\(415\) −61.3545 −3.01177
\(416\) −7.81879 −0.383348
\(417\) 0 0
\(418\) −4.90990 −0.240151
\(419\) 4.44671 0.217236 0.108618 0.994084i \(-0.465357\pi\)
0.108618 + 0.994084i \(0.465357\pi\)
\(420\) 0 0
\(421\) −15.8360 −0.771798 −0.385899 0.922541i \(-0.626109\pi\)
−0.385899 + 0.922541i \(0.626109\pi\)
\(422\) −67.9544 −3.30797
\(423\) 0 0
\(424\) −18.9819 −0.921844
\(425\) 2.05475 0.0996699
\(426\) 0 0
\(427\) −29.5345 −1.42927
\(428\) −34.9518 −1.68946
\(429\) 0 0
\(430\) −49.3672 −2.38070
\(431\) −19.9233 −0.959674 −0.479837 0.877358i \(-0.659304\pi\)
−0.479837 + 0.877358i \(0.659304\pi\)
\(432\) 0 0
\(433\) 5.88331 0.282734 0.141367 0.989957i \(-0.454850\pi\)
0.141367 + 0.989957i \(0.454850\pi\)
\(434\) −53.0733 −2.54760
\(435\) 0 0
\(436\) −1.17658 −0.0563480
\(437\) 5.55926 0.265936
\(438\) 0 0
\(439\) −22.6432 −1.08070 −0.540351 0.841440i \(-0.681708\pi\)
−0.540351 + 0.841440i \(0.681708\pi\)
\(440\) 31.9651 1.52388
\(441\) 0 0
\(442\) 1.39370 0.0662918
\(443\) 19.5044 0.926684 0.463342 0.886180i \(-0.346650\pi\)
0.463342 + 0.886180i \(0.346650\pi\)
\(444\) 0 0
\(445\) 67.2081 3.18597
\(446\) 5.12812 0.242824
\(447\) 0 0
\(448\) 38.4007 1.81426
\(449\) −15.3358 −0.723741 −0.361870 0.932228i \(-0.617862\pi\)
−0.361870 + 0.932228i \(0.617862\pi\)
\(450\) 0 0
\(451\) −10.5764 −0.498021
\(452\) 8.71759 0.410041
\(453\) 0 0
\(454\) 51.4904 2.41656
\(455\) −50.0136 −2.34467
\(456\) 0 0
\(457\) −1.69351 −0.0792192 −0.0396096 0.999215i \(-0.512611\pi\)
−0.0396096 + 0.999215i \(0.512611\pi\)
\(458\) −42.0945 −1.96695
\(459\) 0 0
\(460\) −77.7861 −3.62680
\(461\) 23.4627 1.09277 0.546384 0.837535i \(-0.316004\pi\)
0.546384 + 0.837535i \(0.316004\pi\)
\(462\) 0 0
\(463\) 17.1364 0.796395 0.398197 0.917300i \(-0.369636\pi\)
0.398197 + 0.917300i \(0.369636\pi\)
\(464\) 17.4228 0.808831
\(465\) 0 0
\(466\) −30.5702 −1.41614
\(467\) 24.1942 1.11958 0.559788 0.828636i \(-0.310883\pi\)
0.559788 + 0.828636i \(0.310883\pi\)
\(468\) 0 0
\(469\) 32.4771 1.49965
\(470\) −49.9787 −2.30534
\(471\) 0 0
\(472\) −17.6374 −0.811827
\(473\) −9.36498 −0.430602
\(474\) 0 0
\(475\) −13.0287 −0.597796
\(476\) −2.34734 −0.107590
\(477\) 0 0
\(478\) 2.39589 0.109586
\(479\) 5.10566 0.233284 0.116642 0.993174i \(-0.462787\pi\)
0.116642 + 0.993174i \(0.462787\pi\)
\(480\) 0 0
\(481\) 11.5832 0.528150
\(482\) −30.0005 −1.36648
\(483\) 0 0
\(484\) −28.1109 −1.27777
\(485\) −72.2169 −3.27920
\(486\) 0 0
\(487\) 4.69712 0.212847 0.106423 0.994321i \(-0.466060\pi\)
0.106423 + 0.994321i \(0.466060\pi\)
\(488\) 33.9748 1.53797
\(489\) 0 0
\(490\) 60.3976 2.72849
\(491\) −32.8000 −1.48024 −0.740122 0.672473i \(-0.765232\pi\)
−0.740122 + 0.672473i \(0.765232\pi\)
\(492\) 0 0
\(493\) 1.20219 0.0541441
\(494\) −8.83715 −0.397602
\(495\) 0 0
\(496\) 15.3352 0.688572
\(497\) −50.3959 −2.26056
\(498\) 0 0
\(499\) −18.9668 −0.849071 −0.424536 0.905411i \(-0.639563\pi\)
−0.424536 + 0.905411i \(0.639563\pi\)
\(500\) 105.492 4.71776
\(501\) 0 0
\(502\) 33.6209 1.50058
\(503\) −15.7855 −0.703840 −0.351920 0.936030i \(-0.614471\pi\)
−0.351920 + 0.936030i \(0.614471\pi\)
\(504\) 0 0
\(505\) −37.0818 −1.65012
\(506\) −22.6464 −1.00675
\(507\) 0 0
\(508\) −2.95757 −0.131221
\(509\) −9.35294 −0.414562 −0.207281 0.978281i \(-0.566461\pi\)
−0.207281 + 0.978281i \(0.566461\pi\)
\(510\) 0 0
\(511\) 22.9810 1.01662
\(512\) −26.7648 −1.18285
\(513\) 0 0
\(514\) 13.6050 0.600092
\(515\) 9.39323 0.413915
\(516\) 0 0
\(517\) −9.48098 −0.416973
\(518\) −29.9408 −1.31552
\(519\) 0 0
\(520\) 57.5328 2.52298
\(521\) 11.6796 0.511691 0.255845 0.966718i \(-0.417646\pi\)
0.255845 + 0.966718i \(0.417646\pi\)
\(522\) 0 0
\(523\) −25.0355 −1.09473 −0.547363 0.836895i \(-0.684368\pi\)
−0.547363 + 0.836895i \(0.684368\pi\)
\(524\) −61.8037 −2.69991
\(525\) 0 0
\(526\) −12.3641 −0.539100
\(527\) 1.05815 0.0460938
\(528\) 0 0
\(529\) 2.64150 0.114848
\(530\) −44.7960 −1.94581
\(531\) 0 0
\(532\) 14.8839 0.645300
\(533\) −19.0360 −0.824541
\(534\) 0 0
\(535\) −38.3784 −1.65924
\(536\) −37.3598 −1.61370
\(537\) 0 0
\(538\) 4.05454 0.174804
\(539\) 11.4575 0.493508
\(540\) 0 0
\(541\) 22.3624 0.961433 0.480717 0.876876i \(-0.340377\pi\)
0.480717 + 0.876876i \(0.340377\pi\)
\(542\) −28.0903 −1.20658
\(543\) 0 0
\(544\) −0.402943 −0.0172761
\(545\) −1.29193 −0.0553402
\(546\) 0 0
\(547\) 16.6583 0.712256 0.356128 0.934437i \(-0.384097\pi\)
0.356128 + 0.934437i \(0.384097\pi\)
\(548\) −59.0047 −2.52056
\(549\) 0 0
\(550\) 53.0740 2.26308
\(551\) −7.62282 −0.324743
\(552\) 0 0
\(553\) −5.01732 −0.213358
\(554\) 60.4817 2.56962
\(555\) 0 0
\(556\) 55.7672 2.36506
\(557\) 41.6534 1.76491 0.882456 0.470394i \(-0.155888\pi\)
0.882456 + 0.470394i \(0.155888\pi\)
\(558\) 0 0
\(559\) −16.8557 −0.712920
\(560\) −37.3538 −1.57849
\(561\) 0 0
\(562\) 56.8387 2.39760
\(563\) 42.9853 1.81161 0.905807 0.423691i \(-0.139266\pi\)
0.905807 + 0.423691i \(0.139266\pi\)
\(564\) 0 0
\(565\) 9.57223 0.402707
\(566\) 68.8162 2.89256
\(567\) 0 0
\(568\) 57.9726 2.43247
\(569\) 26.8803 1.12688 0.563440 0.826157i \(-0.309477\pi\)
0.563440 + 0.826157i \(0.309477\pi\)
\(570\) 0 0
\(571\) −21.5622 −0.902351 −0.451176 0.892435i \(-0.648995\pi\)
−0.451176 + 0.892435i \(0.648995\pi\)
\(572\) 23.4566 0.980771
\(573\) 0 0
\(574\) 49.2049 2.05377
\(575\) −60.0933 −2.50607
\(576\) 0 0
\(577\) 42.1498 1.75472 0.877359 0.479835i \(-0.159304\pi\)
0.877359 + 0.479835i \(0.159304\pi\)
\(578\) −40.6584 −1.69116
\(579\) 0 0
\(580\) 106.660 4.42881
\(581\) 54.1486 2.24646
\(582\) 0 0
\(583\) −8.49782 −0.351944
\(584\) −26.4360 −1.09393
\(585\) 0 0
\(586\) −13.5690 −0.560530
\(587\) 18.8112 0.776420 0.388210 0.921571i \(-0.373094\pi\)
0.388210 + 0.921571i \(0.373094\pi\)
\(588\) 0 0
\(589\) −6.70949 −0.276460
\(590\) −41.6230 −1.71359
\(591\) 0 0
\(592\) 8.65121 0.355563
\(593\) 17.1975 0.706218 0.353109 0.935582i \(-0.385125\pi\)
0.353109 + 0.935582i \(0.385125\pi\)
\(594\) 0 0
\(595\) −2.57746 −0.105666
\(596\) 61.9717 2.53846
\(597\) 0 0
\(598\) −40.7604 −1.66682
\(599\) −7.02859 −0.287181 −0.143590 0.989637i \(-0.545865\pi\)
−0.143590 + 0.989637i \(0.545865\pi\)
\(600\) 0 0
\(601\) −32.4567 −1.32394 −0.661968 0.749532i \(-0.730279\pi\)
−0.661968 + 0.749532i \(0.730279\pi\)
\(602\) 43.5692 1.77575
\(603\) 0 0
\(604\) −42.9546 −1.74780
\(605\) −30.8668 −1.25491
\(606\) 0 0
\(607\) 11.6545 0.473040 0.236520 0.971627i \(-0.423993\pi\)
0.236520 + 0.971627i \(0.423993\pi\)
\(608\) 2.55497 0.103618
\(609\) 0 0
\(610\) 80.1781 3.24632
\(611\) −17.0645 −0.690355
\(612\) 0 0
\(613\) −2.82297 −0.114019 −0.0570093 0.998374i \(-0.518156\pi\)
−0.0570093 + 0.998374i \(0.518156\pi\)
\(614\) −42.5698 −1.71798
\(615\) 0 0
\(616\) −28.2109 −1.13665
\(617\) 23.5462 0.947936 0.473968 0.880542i \(-0.342821\pi\)
0.473968 + 0.880542i \(0.342821\pi\)
\(618\) 0 0
\(619\) 38.2730 1.53832 0.769162 0.639054i \(-0.220674\pi\)
0.769162 + 0.639054i \(0.220674\pi\)
\(620\) 93.8803 3.77032
\(621\) 0 0
\(622\) 59.5217 2.38660
\(623\) −59.3147 −2.37640
\(624\) 0 0
\(625\) 56.4977 2.25991
\(626\) −70.3224 −2.81065
\(627\) 0 0
\(628\) −39.4854 −1.57564
\(629\) 0.596945 0.0238018
\(630\) 0 0
\(631\) 14.2574 0.567576 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(632\) 5.77164 0.229584
\(633\) 0 0
\(634\) −54.5928 −2.16816
\(635\) −3.24752 −0.128874
\(636\) 0 0
\(637\) 20.6219 0.817068
\(638\) 31.0526 1.22938
\(639\) 0 0
\(640\) −85.1314 −3.36512
\(641\) −31.3172 −1.23696 −0.618478 0.785802i \(-0.712250\pi\)
−0.618478 + 0.785802i \(0.712250\pi\)
\(642\) 0 0
\(643\) 50.4477 1.98946 0.994732 0.102512i \(-0.0326880\pi\)
0.994732 + 0.102512i \(0.0326880\pi\)
\(644\) 68.6505 2.70521
\(645\) 0 0
\(646\) −0.455425 −0.0179185
\(647\) 2.82116 0.110911 0.0554555 0.998461i \(-0.482339\pi\)
0.0554555 + 0.998461i \(0.482339\pi\)
\(648\) 0 0
\(649\) −7.89590 −0.309941
\(650\) 95.5260 3.74684
\(651\) 0 0
\(652\) 71.1344 2.78584
\(653\) −43.4096 −1.69875 −0.849374 0.527792i \(-0.823020\pi\)
−0.849374 + 0.527792i \(0.823020\pi\)
\(654\) 0 0
\(655\) −67.8627 −2.65162
\(656\) −14.2175 −0.555099
\(657\) 0 0
\(658\) 44.1089 1.71954
\(659\) 36.4540 1.42005 0.710024 0.704178i \(-0.248684\pi\)
0.710024 + 0.704178i \(0.248684\pi\)
\(660\) 0 0
\(661\) 13.0915 0.509201 0.254600 0.967046i \(-0.418056\pi\)
0.254600 + 0.967046i \(0.418056\pi\)
\(662\) 55.7589 2.16713
\(663\) 0 0
\(664\) −62.2895 −2.41730
\(665\) 16.3431 0.633758
\(666\) 0 0
\(667\) −35.1594 −1.36138
\(668\) 5.75366 0.222616
\(669\) 0 0
\(670\) −88.1663 −3.40616
\(671\) 15.2098 0.587169
\(672\) 0 0
\(673\) 3.94420 0.152038 0.0760188 0.997106i \(-0.475779\pi\)
0.0760188 + 0.997106i \(0.475779\pi\)
\(674\) 77.7913 2.99641
\(675\) 0 0
\(676\) −6.40517 −0.246353
\(677\) 34.7858 1.33693 0.668464 0.743744i \(-0.266952\pi\)
0.668464 + 0.743744i \(0.266952\pi\)
\(678\) 0 0
\(679\) 63.7353 2.44594
\(680\) 2.96497 0.113701
\(681\) 0 0
\(682\) 27.3320 1.04660
\(683\) −32.7697 −1.25390 −0.626949 0.779060i \(-0.715697\pi\)
−0.626949 + 0.779060i \(0.715697\pi\)
\(684\) 0 0
\(685\) −64.7893 −2.47547
\(686\) 7.48561 0.285802
\(687\) 0 0
\(688\) −12.5891 −0.479954
\(689\) −15.2949 −0.582690
\(690\) 0 0
\(691\) −27.6403 −1.05149 −0.525744 0.850643i \(-0.676213\pi\)
−0.525744 + 0.850643i \(0.676213\pi\)
\(692\) −59.9847 −2.28027
\(693\) 0 0
\(694\) −20.6522 −0.783948
\(695\) 61.2345 2.32276
\(696\) 0 0
\(697\) −0.981025 −0.0371590
\(698\) −49.2712 −1.86494
\(699\) 0 0
\(700\) −160.889 −6.08103
\(701\) −17.8186 −0.673000 −0.336500 0.941683i \(-0.609243\pi\)
−0.336500 + 0.941683i \(0.609243\pi\)
\(702\) 0 0
\(703\) −3.78509 −0.142757
\(704\) −19.7758 −0.745327
\(705\) 0 0
\(706\) 67.4506 2.53854
\(707\) 32.7267 1.23081
\(708\) 0 0
\(709\) 28.5417 1.07190 0.535952 0.844248i \(-0.319953\pi\)
0.535952 + 0.844248i \(0.319953\pi\)
\(710\) 136.811 5.13443
\(711\) 0 0
\(712\) 68.2323 2.55711
\(713\) −30.9468 −1.15897
\(714\) 0 0
\(715\) 25.7563 0.963229
\(716\) 33.3793 1.24744
\(717\) 0 0
\(718\) −32.4913 −1.21256
\(719\) 16.9208 0.631040 0.315520 0.948919i \(-0.397821\pi\)
0.315520 + 0.948919i \(0.397821\pi\)
\(720\) 0 0
\(721\) −8.29004 −0.308737
\(722\) −42.6342 −1.58668
\(723\) 0 0
\(724\) 39.3570 1.46269
\(725\) 82.3996 3.06024
\(726\) 0 0
\(727\) −3.88604 −0.144125 −0.0720627 0.997400i \(-0.522958\pi\)
−0.0720627 + 0.997400i \(0.522958\pi\)
\(728\) −50.7758 −1.88188
\(729\) 0 0
\(730\) −62.3872 −2.30905
\(731\) −0.868663 −0.0321287
\(732\) 0 0
\(733\) 11.3841 0.420480 0.210240 0.977650i \(-0.432575\pi\)
0.210240 + 0.977650i \(0.432575\pi\)
\(734\) −61.8109 −2.28148
\(735\) 0 0
\(736\) 11.7845 0.434383
\(737\) −16.7252 −0.616081
\(738\) 0 0
\(739\) 31.2138 1.14822 0.574109 0.818779i \(-0.305349\pi\)
0.574109 + 0.818779i \(0.305349\pi\)
\(740\) 52.9616 1.94691
\(741\) 0 0
\(742\) 39.5349 1.45137
\(743\) −29.8800 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(744\) 0 0
\(745\) 68.0472 2.49306
\(746\) 85.2368 3.12074
\(747\) 0 0
\(748\) 1.20884 0.0441997
\(749\) 33.8710 1.23762
\(750\) 0 0
\(751\) 0.548324 0.0200086 0.0100043 0.999950i \(-0.496815\pi\)
0.0100043 + 0.999950i \(0.496815\pi\)
\(752\) −12.7450 −0.464763
\(753\) 0 0
\(754\) 55.8904 2.03541
\(755\) −47.1657 −1.71654
\(756\) 0 0
\(757\) −38.9526 −1.41576 −0.707878 0.706335i \(-0.750347\pi\)
−0.707878 + 0.706335i \(0.750347\pi\)
\(758\) −6.05868 −0.220061
\(759\) 0 0
\(760\) −18.8002 −0.681954
\(761\) 6.47569 0.234743 0.117372 0.993088i \(-0.462553\pi\)
0.117372 + 0.993088i \(0.462553\pi\)
\(762\) 0 0
\(763\) 1.14020 0.0412779
\(764\) 79.2477 2.86708
\(765\) 0 0
\(766\) −32.9690 −1.19122
\(767\) −14.2116 −0.513149
\(768\) 0 0
\(769\) 9.89337 0.356764 0.178382 0.983961i \(-0.442914\pi\)
0.178382 + 0.983961i \(0.442914\pi\)
\(770\) −66.5757 −2.39922
\(771\) 0 0
\(772\) −20.7611 −0.747210
\(773\) 12.2179 0.439447 0.219723 0.975562i \(-0.429485\pi\)
0.219723 + 0.975562i \(0.429485\pi\)
\(774\) 0 0
\(775\) 72.5268 2.60524
\(776\) −73.3175 −2.63195
\(777\) 0 0
\(778\) −55.2722 −1.98160
\(779\) 6.22045 0.222871
\(780\) 0 0
\(781\) 25.9531 0.928676
\(782\) −2.10060 −0.0751173
\(783\) 0 0
\(784\) 15.4019 0.550069
\(785\) −43.3565 −1.54746
\(786\) 0 0
\(787\) −32.8255 −1.17010 −0.585052 0.810996i \(-0.698926\pi\)
−0.585052 + 0.810996i \(0.698926\pi\)
\(788\) 61.7122 2.19841
\(789\) 0 0
\(790\) 13.6207 0.484601
\(791\) −8.44801 −0.300377
\(792\) 0 0
\(793\) 27.3756 0.972137
\(794\) 38.7258 1.37433
\(795\) 0 0
\(796\) 77.1203 2.73346
\(797\) 52.9963 1.87723 0.938613 0.344972i \(-0.112111\pi\)
0.938613 + 0.344972i \(0.112111\pi\)
\(798\) 0 0
\(799\) −0.879423 −0.0311117
\(800\) −27.6182 −0.976449
\(801\) 0 0
\(802\) −33.3068 −1.17611
\(803\) −11.8349 −0.417644
\(804\) 0 0
\(805\) 75.3808 2.65682
\(806\) 49.1939 1.73278
\(807\) 0 0
\(808\) −37.6469 −1.32441
\(809\) 8.64210 0.303840 0.151920 0.988393i \(-0.451454\pi\)
0.151920 + 0.988393i \(0.451454\pi\)
\(810\) 0 0
\(811\) 15.2237 0.534575 0.267287 0.963617i \(-0.413873\pi\)
0.267287 + 0.963617i \(0.413873\pi\)
\(812\) −94.1331 −3.30342
\(813\) 0 0
\(814\) 15.4191 0.540438
\(815\) 78.1082 2.73601
\(816\) 0 0
\(817\) 5.50798 0.192700
\(818\) −18.6812 −0.653172
\(819\) 0 0
\(820\) −87.0376 −3.03948
\(821\) −12.0526 −0.420637 −0.210318 0.977633i \(-0.567450\pi\)
−0.210318 + 0.977633i \(0.567450\pi\)
\(822\) 0 0
\(823\) −48.1644 −1.67891 −0.839453 0.543432i \(-0.817124\pi\)
−0.839453 + 0.543432i \(0.817124\pi\)
\(824\) 9.53639 0.332216
\(825\) 0 0
\(826\) 36.7345 1.27816
\(827\) −1.62044 −0.0563481 −0.0281740 0.999603i \(-0.508969\pi\)
−0.0281740 + 0.999603i \(0.508969\pi\)
\(828\) 0 0
\(829\) 29.6369 1.02933 0.514666 0.857391i \(-0.327916\pi\)
0.514666 + 0.857391i \(0.327916\pi\)
\(830\) −146.999 −5.10240
\(831\) 0 0
\(832\) −35.5937 −1.23399
\(833\) 1.06275 0.0368222
\(834\) 0 0
\(835\) 6.31773 0.218634
\(836\) −7.66499 −0.265099
\(837\) 0 0
\(838\) 10.6538 0.368031
\(839\) 39.2976 1.35671 0.678353 0.734737i \(-0.262694\pi\)
0.678353 + 0.734737i \(0.262694\pi\)
\(840\) 0 0
\(841\) 19.2104 0.662428
\(842\) −37.9413 −1.30754
\(843\) 0 0
\(844\) −106.086 −3.65162
\(845\) −7.03311 −0.241946
\(846\) 0 0
\(847\) 27.2416 0.936033
\(848\) −11.4234 −0.392280
\(849\) 0 0
\(850\) 4.92296 0.168856
\(851\) −17.4583 −0.598463
\(852\) 0 0
\(853\) −3.53146 −0.120915 −0.0604574 0.998171i \(-0.519256\pi\)
−0.0604574 + 0.998171i \(0.519256\pi\)
\(854\) −70.7615 −2.42141
\(855\) 0 0
\(856\) −38.9633 −1.33174
\(857\) 22.3714 0.764191 0.382096 0.924123i \(-0.375203\pi\)
0.382096 + 0.924123i \(0.375203\pi\)
\(858\) 0 0
\(859\) 6.24869 0.213203 0.106601 0.994302i \(-0.466003\pi\)
0.106601 + 0.994302i \(0.466003\pi\)
\(860\) −77.0687 −2.62802
\(861\) 0 0
\(862\) −47.7342 −1.62583
\(863\) 23.4800 0.799270 0.399635 0.916674i \(-0.369137\pi\)
0.399635 + 0.916674i \(0.369137\pi\)
\(864\) 0 0
\(865\) −65.8653 −2.23949
\(866\) 14.0958 0.478994
\(867\) 0 0
\(868\) −82.8545 −2.81226
\(869\) 2.58384 0.0876509
\(870\) 0 0
\(871\) −30.1031 −1.02000
\(872\) −1.31162 −0.0444170
\(873\) 0 0
\(874\) 13.3194 0.450535
\(875\) −102.230 −3.45601
\(876\) 0 0
\(877\) −22.1035 −0.746382 −0.373191 0.927755i \(-0.621736\pi\)
−0.373191 + 0.927755i \(0.621736\pi\)
\(878\) −54.2507 −1.83087
\(879\) 0 0
\(880\) 19.2367 0.648468
\(881\) 18.1078 0.610068 0.305034 0.952341i \(-0.401332\pi\)
0.305034 + 0.952341i \(0.401332\pi\)
\(882\) 0 0
\(883\) 1.96256 0.0660455 0.0330227 0.999455i \(-0.489487\pi\)
0.0330227 + 0.999455i \(0.489487\pi\)
\(884\) 2.17576 0.0731786
\(885\) 0 0
\(886\) 46.7306 1.56994
\(887\) −48.6192 −1.63247 −0.816236 0.577718i \(-0.803943\pi\)
−0.816236 + 0.577718i \(0.803943\pi\)
\(888\) 0 0
\(889\) 2.86611 0.0961262
\(890\) 161.023 5.39751
\(891\) 0 0
\(892\) 8.00568 0.268050
\(893\) 5.57621 0.186601
\(894\) 0 0
\(895\) 36.6517 1.22513
\(896\) 75.1331 2.51002
\(897\) 0 0
\(898\) −36.7429 −1.22613
\(899\) 42.4340 1.41525
\(900\) 0 0
\(901\) −0.788228 −0.0262597
\(902\) −25.3398 −0.843723
\(903\) 0 0
\(904\) 9.71812 0.323220
\(905\) 43.2155 1.43653
\(906\) 0 0
\(907\) 17.3975 0.577675 0.288837 0.957378i \(-0.406731\pi\)
0.288837 + 0.957378i \(0.406731\pi\)
\(908\) 80.3832 2.66761
\(909\) 0 0
\(910\) −119.827 −3.97223
\(911\) 9.19217 0.304550 0.152275 0.988338i \(-0.451340\pi\)
0.152275 + 0.988338i \(0.451340\pi\)
\(912\) 0 0
\(913\) −27.8857 −0.922883
\(914\) −4.05748 −0.134209
\(915\) 0 0
\(916\) −65.7150 −2.17129
\(917\) 59.8925 1.97783
\(918\) 0 0
\(919\) 47.2057 1.55717 0.778587 0.627537i \(-0.215937\pi\)
0.778587 + 0.627537i \(0.215937\pi\)
\(920\) −86.7137 −2.85887
\(921\) 0 0
\(922\) 56.2142 1.85132
\(923\) 46.7121 1.53755
\(924\) 0 0
\(925\) 40.9153 1.34529
\(926\) 41.0569 1.34921
\(927\) 0 0
\(928\) −16.1588 −0.530440
\(929\) 15.7119 0.515490 0.257745 0.966213i \(-0.417021\pi\)
0.257745 + 0.966213i \(0.417021\pi\)
\(930\) 0 0
\(931\) −6.73867 −0.220851
\(932\) −47.7242 −1.56326
\(933\) 0 0
\(934\) 57.9668 1.89673
\(935\) 1.32736 0.0434092
\(936\) 0 0
\(937\) −39.2075 −1.28085 −0.640427 0.768019i \(-0.721242\pi\)
−0.640427 + 0.768019i \(0.721242\pi\)
\(938\) 77.8116 2.54064
\(939\) 0 0
\(940\) −78.0233 −2.54484
\(941\) 13.4619 0.438844 0.219422 0.975630i \(-0.429583\pi\)
0.219422 + 0.975630i \(0.429583\pi\)
\(942\) 0 0
\(943\) 28.6912 0.934312
\(944\) −10.6142 −0.345464
\(945\) 0 0
\(946\) −22.4375 −0.729506
\(947\) −9.05254 −0.294168 −0.147084 0.989124i \(-0.546989\pi\)
−0.147084 + 0.989124i \(0.546989\pi\)
\(948\) 0 0
\(949\) −21.3012 −0.691465
\(950\) −31.2153 −1.01276
\(951\) 0 0
\(952\) −2.61675 −0.0848092
\(953\) −29.5658 −0.957729 −0.478864 0.877889i \(-0.658951\pi\)
−0.478864 + 0.877889i \(0.658951\pi\)
\(954\) 0 0
\(955\) 87.0169 2.81580
\(956\) 3.74030 0.120970
\(957\) 0 0
\(958\) 12.2326 0.395218
\(959\) 57.1801 1.84644
\(960\) 0 0
\(961\) 6.34978 0.204831
\(962\) 27.7522 0.894767
\(963\) 0 0
\(964\) −46.8347 −1.50844
\(965\) −22.7965 −0.733845
\(966\) 0 0
\(967\) 20.9682 0.674290 0.337145 0.941453i \(-0.390539\pi\)
0.337145 + 0.941453i \(0.390539\pi\)
\(968\) −31.3372 −1.00722
\(969\) 0 0
\(970\) −173.024 −5.55547
\(971\) 14.5417 0.466664 0.233332 0.972397i \(-0.425037\pi\)
0.233332 + 0.972397i \(0.425037\pi\)
\(972\) 0 0
\(973\) −54.0427 −1.73253
\(974\) 11.2538 0.360595
\(975\) 0 0
\(976\) 20.4461 0.654464
\(977\) 42.9015 1.37254 0.686270 0.727347i \(-0.259247\pi\)
0.686270 + 0.727347i \(0.259247\pi\)
\(978\) 0 0
\(979\) 30.5462 0.976261
\(980\) 94.2886 3.01194
\(981\) 0 0
\(982\) −78.5853 −2.50776
\(983\) 8.10974 0.258661 0.129330 0.991602i \(-0.458717\pi\)
0.129330 + 0.991602i \(0.458717\pi\)
\(984\) 0 0
\(985\) 67.7623 2.15909
\(986\) 2.88033 0.0917283
\(987\) 0 0
\(988\) −13.7960 −0.438908
\(989\) 25.4050 0.807831
\(990\) 0 0
\(991\) 39.5827 1.25738 0.628692 0.777654i \(-0.283591\pi\)
0.628692 + 0.777654i \(0.283591\pi\)
\(992\) −14.2228 −0.451573
\(993\) 0 0
\(994\) −120.743 −3.82974
\(995\) 84.6809 2.68456
\(996\) 0 0
\(997\) 35.5573 1.12611 0.563056 0.826419i \(-0.309626\pi\)
0.563056 + 0.826419i \(0.309626\pi\)
\(998\) −45.4425 −1.43846
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2151.2.a.k.1.17 yes 20
3.2 odd 2 2151.2.a.j.1.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.4 20 3.2 odd 2
2151.2.a.k.1.17 yes 20 1.1 even 1 trivial